# Abelian-extensible automorphism-invariant subgroup

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

Suppose $G$ is an abelian group and $H$ is a subgroup of $G$. We say that $H$ is an abelian-extensible automorphism-invariant subgroup of $G$ if, for every abelian-extensible automorphism $\sigma$ of $G$, we have $\sigma(H) = H$.

## Formalisms

### Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property
Function restriction expression $H$ is a fully invariant subgroup of $G$ if ... This means that full invariance is ... Additional comments
abelian-extensible automorphism $\to$ function every abelian-extensible automorphism of $G$ sends every element of $H$ to within $H$ the invariance property for abelian-extensible automorphisms
abelian-extensible automorphism $\to$ endomorphism every abelian-extensible automorphism of $G$ restricts to an endomorphism of $H$ the endo-invariance property for abelian-extensible automorphisms; i.e., it is the invariance property for abelian-extensible automorphism, which is a property stronger than the property of being an endomorphism
abelian-extensible automorphism $\to$ automorphism every abelian-extensible automorphism of $G$ restricts to an automorphism of $H$ the auto-invariance property for abelian-extensible automorphisms; i.e., it is the invariance property for abelian-extensible automorphism, which is a group-closed property of automorphisms

## Relation with other properties

### Stronger properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions